Dirichlet series
| L(χ,s) = 1 | + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)5-s − 8-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + 13-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s − 20-s + 22-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + (0.5 − 0.866i)26-s + ⋯ |
| L(s,χ) = 1 | + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)5-s − 8-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + 13-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s − 20-s + 22-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + (0.5 − 0.866i)26-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(\chi,s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr
=\mathstrut & (-0.0633 - 0.997i)\, \Lambda(\overline{\chi},1-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s,\chi)=\mathstrut & 21 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr
=\mathstrut & (-0.0633 - 0.997i)\, \Lambda(1-s,\overline{\chi})
\end{aligned}
\]
Invariants
| \( d \) | = | \(1\) |
| \( N \) | = | \(21\) = \(3 \cdot 7\) |
| \( \varepsilon \) | = | $-0.0633 - 0.997i$ |
| motivic weight | = | \(0\) |
| character | : | $\chi_{21} (11, \cdot )$ |
| Sato-Tate | : | $\mu(6)$ |
| primitive | : | yes |
| self-dual | : | no |
| analytic rank | = | 0 |
| Selberg data | = | $(1,\ 21,\ (1:\ ),\ -0.0633 - 0.997i)$ |
| $L(\chi,\frac{1}{2})$ | $\approx$ | $1.119373288 - 1.192660627i$ |
| $L(\frac12,\chi)$ | $\approx$ | $1.119373288 - 1.192660627i$ |
| $L(\chi,1)$ | $\approx$ | 1.141608742 - 0.7593774607i |
| $L(1,\chi)$ | $\approx$ | 1.141608742 - 0.7593774607i |
Euler product
\[\begin{aligned}
L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}
\end{aligned}\]
\[\begin{aligned}
L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}
\end{aligned}\]